Can I install a ceiling-mounted exhaust fan (350 CFM) over my kitchen island, with 48-inch gas range, rather than a hood? If so, can you recommend an exhaust fan?
Sure. You may install an exhaust fan instead of hood. The fan should be positioned right above your stove and make sure that there is no obstacle between fan and stove. Also proper ventilation is necessary to let air in.
A 48 inch gas range sounds large to me and 350 CFM may not be enough. Many people have up to 1000 CFM for smaller ranges and cooktops.
A hood will help collect the cooking fumes while the aperture of an exhaust fan is not as large.
Real kitchen hoods have grease collection mechanisms and are designed to be easy to clean. An exhaust fan will not.
Don't forget to provide a way to replace the air you remove or it will be as effective as a clogged vacuum cleaner.